We consider the Cauchy problem for a strictly hyperbolic $2times 2$ system of conservation laws in one space dimension $u_t+[F(u)]_x=0, u(0,x)=bar u(x),$ which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If $r_i(u), i=1,2,$ denotes the $i$-th right eigenvector of $DF(u)$ and $lambda_i(u)$ the corresponding eigenvalue, then the set ${u: nabla lambda_i cdot r_i (u) = 0}$ is a smooth curve in the $u$-plane that is transversal to the vector field $r_i(u)$. Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.For such systems we prove the existence of a closed domain $mathcal{D} subset L^1,$ containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup $S:mathcal{D} times [0,+infty)rightarrow mathcal{D}$ with the following properties. Each trajectory $t mapsto S_t bar u$ of $S$ is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution $u= u(t,x)$ of (1) exists for $t in [0,T],$ then it coincides with the trajectory of $S$, i.e. $u(t,cdot) = S_t bar u. This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem with small initial data, for systems satysfying the above assumption.